94 ANALYTICAL MECHANICS
sin €i and sin e2 approach, ei and <$* respectively. Therefore ihe normal acceleration at P is
= limit — v 7
t=Q L IJ
de «-*-—*, (XII)
where 0= ei+e2 is the total change in the direction of the velocity in going from Pi to P2. Since the direction of the velocity coincides with that of the tangent, 0 is the rate at which the directions of the tangent and the normal change. But the rate at which the normal changes its direction equals the angular velocity of the particle about the center of curvature. Therefore if p denotes the radius of curvature at P, we have
§4 _ Ibyvii,
and fn=~~ (XIII)
The negative sign in (XIII) shows that/n and p are measured in opposite directions. Since p is measured from the center of curvature, fn must be directed towards the center of curvature. Therefore the total acceleration is always directed towards the concave side of the path.
The following are the principal results obtained in this section and the conclusions to be drawn from them.
(a) The magnitude of the tangential acceleration is v;
fr = v.
(6) The normal acceleration is directed towards the center
v2 of curvature and has — for its magnitude;
* See Appendix AVI.